# Non-Diagonal Fall-Back Model

#### Carl James Schwarz

#### 2021-10-24

# 1 Location of vignette source and code.

Because of the length of time needed to run the vignettes, only static vignettes have been included with this package.

The original of the vignettes and the code can be obtained from the GitHub site at https://github.com/cschwarz-stat-sfu-ca/BTSPAS

# 2 Introduction

## 2.1 Experimental set-up

This case represents a generalization of the non-diagonal case considered in a separate vignette. Now we allow some fish (marked and unmarked) to approach the second trap, but fall back and never pass the trap. Schwarz and Bonner (2011) considered this model to estimate the number of steelhead that passed upstream of Moricetown Canyon.

The experimental setup is the same as the non-diagonal case. Consider an experiment to estimate the number of outgoing smolts on a small river. The run of smolts extends over several weeks. As smolts migrate, they are captured and marked with individually numbered tags and released at the first capture location using, for example, a fishwheel. The migration continues, and a second fishwheel takes a second sample several kilometers down stream. At the second fishwheel, the captures consist of a mixture of marked (from the first fishwheel) and unmarked fish.

The efficiency of the fishwheels varies over time in response to stream flow, run size passing the wheel and other uncontrollable events. So it is unlikely that the capture probabilities are equal over time at either location, i.e. are heterogeneous over time.

We suppose that we can temporally stratify the data into, for example, weeks, where the capture-probabilities are (mostly) homogeneous at each wheel in each week.

But now, we allow tagged animals to be captured in several recovery strata. For example, suppose that in each julian week \(j\), \(n1[j]\) fish are marked and released above the rotary screw trap. Of these, \(m2[j,j]\) are recaptured in julian week \(j\); \(m2[j,j+1]\) are recaptured in julian week \(j+1\); \(m2[j,j+2]\) are recaptured in julian week \(j+2\) and so on.

At the same time, \(u2[j]\) unmarked fish are captured at the screw trap.

This implies that the data can be structured as a non-diagonal array similar to:

    Recovery Stratum
                   tagged    rs1      rs2     rs3 ...rs4                 rsk  rs(k+1)
    Marking   ms1    n1[1]  m2[1,1] m2[1,2] m2[1,3] m2[1,4]      0  ...   0      0
    Stratum   ms2    n1[2]   0      m2[2,2] m2[2,3] m2[2,4] .... 0  ...   0      0
              ms3    n1[3]   0       0      m2[3,3] m2[3,4] ...  0  ...   0      0
              ...
              msk    n1[k]   0       0      0  ...  0            0    m2[k,k] m2[k,k+1]
    Newly
    Untagged               u2[1]   u2[2]   u2[3]  ...                 u2[k]   u2[k,k+1]
    captured

Here the tagging and recapture events have been stratified in to \(k\) temporal strata. Marked fish from one stratum tend to spread out and are recaptured over multiple strata. Several additional recovery strata are needed at the end of the experiment to fully capture the final release stratum.

Because the lower diagonal of the recovery matrix is zero, the data can be entered in a shorthand fashion by showing the recoveries in the same stratum as release, the next stratum, etc, up to a maximum number of recovery strata per release.

## 2.2 Fall-back information

This information is obtained by also marking radio-tagged fish whose ultimate fate (i.e. did they pass the second trap nor not) can be determined. We measure:

- \(marked\_available\_n\) representing the number of radio-tagged fish.
- \(marked\_available\_x\) representing the number of radio tagged fish that PASSED the second trap.The \(n\) and \(x\) are modelled using a binomial distribution for information on the fraction of tagged fish that DO NOT fall back, i.e. are available at the second trap. For example, if \(n=66\) and \(x=40\), then you estimate that about \(40/66=61\)% of tagged and untagged fish pass the second trap and that \(39\)% of fish fall back never to pass the second trap.

Notice we don’t really care about unmarked fish that fall back as we only estimate the number of unmarked fish that pass the second trap, which by definition exclude those fish that never make it to second trap. We need to worry about marked fish that never make it to the second trap because the fish that fall back will lead to underestimates of the trap-efficiency and over-estimates of unmarked fish that pass the second trap.

This model could also be used for mortality between the marking and recovery trap.

## 2.3 Fixing values of \(p\) or using covariates.

Refer to the vignette on the Diagonal Case for information about fixing values of \(p\) or modelling \(p\) using covariates such a stream flow or smoothing \(p\) using a temporal spline.

# 3 Example of non-diagonal model with fall-back.

## 3.1 Reading in the data

Here is an example of some raw data that is read in:

    demo.data.csv <- textConnection("
    jweek,n1,    X0,X1 ,X2 ,X3,X4,X5,X6,X7
    29 ,  1 ,    0 , 0 , 0 ,0 ,0 ,0 ,0 ,0
    30 , 35 ,    0 , 5 , 7 ,2 ,0 ,0 ,0 ,0
    31 ,186 ,    1 ,35 ,11 ,4 ,0 ,0 ,0 ,0
    32 ,292 ,    9 ,33 ,16 ,6 ,0 ,0 ,0 ,0
    33 ,460 ,    6 ,41 ,16 ,9 ,3 ,0 ,2 ,1
    34 ,397 ,    4 ,44 , 7 ,5 ,1 ,1 ,0 ,1
    35 ,492 ,    7 ,31 ,12 ,1 ,4 ,1 ,1 ,0
    36 ,151 ,    3 , 6 , 2 ,1 ,1 ,0 ,0 ,0
    37 ,130 ,    3 , 2 , 2 ,0 ,0 ,1 ,0 ,0
    38 ,557 ,    8 ,27 ,11 ,2 ,5 ,0 ,0 ,0
    39 , 46 ,    0 , 7 , 0 ,0 ,0 ,0 ,0 ,0
    40 ,143 ,   14 , 6 , 3 ,0 ,0 ,0 ,0 ,0
    41 , 26 ,    2 , 1 , 0 ,0 ,0 ,0 ,0 ,0")

    # Read data
    demo.data <- read.csv(demo.data.csv, header=TRUE, as.is=TRUE, strip.white=TRUE)

    print(demo.data)
    #>    jweek  n1 X0 X1 X2 X3 X4 X5 X6 X7
    #> 1     29   1  0  0  0  0  0  0  0  0
    #> 2     30  35  0  5  7  2  0  0  0  0
    #> 3     31 186  1 35 11  4  0  0  0  0
    #> 4     32 292  9 33 16  6  0  0  0  0
    #> 5     33 460  6 41 16  9  3  0  2  1
    #> 6     34 397  4 44  7  5  1  1  0  1
    #> 7     35 492  7 31 12  1  4  1  1  0
    #> 8     36 151  3  6  2  1  1  0  0  0
    #> 9     37 130  3  2  2  0  0  1  0  0
    #> 10    38 557  8 27 11  2  5  0  0  0
    #> 11    39  46  0  7  0  0  0  0  0  0
    #> 12    40 143 14  6  3  0  0  0  0  0
    #> 13    41  26  2  1  0  0  0  0  0  0

There are 13 release strata. In the first release stratum, a total of 1 fish were tagged and released. No recoveries occurred.

Because the recoveries take place in more strata than releases, the \(u2\) vector is read in separately. Note that is must be sufficiently long to account for the number of releases plus potential movement:

    demo.data.u2 <- c(  2,  65,  325,  873,  976,  761,  869,  473,  332,  197,
                      177, 282,   82,  100)

We also separate out the recoveries \(m2\) into a matrix

    demo.data.m2 <- as.matrix(demo.data[,c("X0","X1","X2","X3","X4","X5","X6","X7")])

A separate radio-telemetry study found that of 66 fish released, 40 passed the second trap:

    demo.mark.available.n <- 66
    demo.mark.available.x <- 40

## 3.2 Fitting the BTSPAS non-diagonal model with fall-back model with a non-parametric movement distribution.

Schwarz and Bonner (2011) extended Bonner and Schwarz (2011) with a model with the following features.

- Non-parametric distribution of the distribution of times between release and availability at the second trap.
- A spline is used to smooth the total number of unmarked fish presenting themselves at the second trap over the strata
- A hierarchical model for the capture-probabilities is assumed where individual stratum capture probabilities are assumed to vary around a common mean.
- A binomial distribution is assumed for the number of marked fish that do not fall back and pass the second trap to estimate the trap efficiency.

The model also allows the user to use covariates to explain some of the variation in the capture probabilities in much the same way as the diagonal case.

The \(BTSPAS\) package also has additional features and options:

- if \(u2\) is missing for any stratum, the program will use the spline to interpolate for the number of unmarked fish in the population missing stratum.
- if \(n1\) and the entire corresponding row of \(m2\) are 0, the program will use the hierarchical model to interpolate the capture probabilities for the missing strata because no information is available from 0 fish released.
- the program allows you specify break points in the underlying spline to account for external events.
- sometimes bad thing happen. The vector \(bad.m2\) indicates which julian weeks something went wrong. In the above example, the number of recoveries in julian week 41 is far below expectations and leads to impossible Petersen estimate for julian week 41. Similarly, the vector \(bad.u2\) indicates which julian weeks, the number of unmarked fish is suspect. In both cases, the suspect values of \(n1\) and \(m2\) are set to 0, and the suspect values of \(u2\) are set to missing. Alternatively, the user can set the \(n1\) and \(m2\) values to 0, and set the suspect \(u2\) values to missing in the data input directly. I arbitrarily chose the third julian week to demonstrate this feature.

The \(BTSPAS\) function also allows you specify

- The prefix is used to identify the output files for this run.
- The title is used to title the output.
- Various parameters to control the Bayesian MCMC phase of model fitting. Please contact us for help in setting these if problem arise.

We already read in the data above. Here we set the rest of the parameters. Don’t forget to set the working directory as appropriate

    library("BTSPAS")
    demo.prefix <- "FB-"
    demo.title  <- "Fall-back demo"

    demo.jump.after <- NULL

    ## Identify spurious values in n1, m2, and u2 that should be set to 0 or missing as needed.
    demo.bad.n1     <- c()     # list sample times of bad n1 values
    demo.bad.m2     <- c()     # list sample times of bad m2 values
    demo.bad.u2     <- c()     # list sample times of bad u2 values

    ## Fix capture probabilities for strata when traps not operated
    demo.logitP.fixed <- NULL
    demo.logitP.fixed.values <- rep(-10,length(demo.logitP.fixed))

    demo.fit <- TimeStratPetersenNonDiagErrorNPMarkAvail_fit(
                      title=      demo.title,
                      prefix=     demo.prefix,
                      time=       demo.data$jweek[1]:(demo.data$jweek[1]+length(demo.data.u2)-1),
                      n1=         demo.data$n1,
                      m2=         demo.data.m2,
                      u2=         demo.data.u2,
                      jump.after= demo.jump.after,
                      bad.n1=     demo.bad.n1,
                      bad.m2=     demo.bad.m2,
                      bad.u2=     demo.bad.u2,
                      logitP.fixed=demo.logitP.fixed,
                      logitP.fixed.values=demo.logitP.fixed.values,
                      marked_available_n=demo.mark.available.n,
                      marked_available_x=demo.mark.available.x,  # 40/66 fish did NOT fall back
                      debug=TRUE,
                      save.output.to.files=FALSE)
    #>
    #>
    #> *** Start of call to JAGS
    #> Working directory:  /Users/cschwarz/Dropbox/SPAS-Bayesian/BTSPAS/vignettes
    #> Initial seed for JAGS set to: 772306
    #> Random number seed for chain  661223
    #> Random number seed for chain  720272
    #> Random number seed for chain  910532
    #> Compiling model graph
    #>    Resolving undeclared variables
    #>    Allocating nodes
    #> Graph information:
    #>    Observed stochastic nodes: 27
    #>    Unobserved stochastic nodes: 141
    #>    Total graph size: 1079
    #>
    #> Initializing model
    #>
    #>
      |
      |                                                  |   0%
      |
      |++                                                |   4%
      |
      |++++                                              |   8%
      |
      |++++++                                            |  12%
      |
      |++++++++                                          |  16%
      |
      |++++++++++                                        |  20%
      |
      |++++++++++++                                      |  24%
      |
      |++++++++++++++                                    |  28%
      |
      |++++++++++++++++                                  |  32%
      |
      |++++++++++++++++++                                |  36%
      |
      |++++++++++++++++++++                              |  40%
      |
      |++++++++++++++++++++++                            |  44%
      |
      |++++++++++++++++++++++++                          |  48%
      |
      |++++++++++++++++++++++++++                        |  52%
      |
      |++++++++++++++++++++++++++++                      |  56%
      |
      |++++++++++++++++++++++++++++++                    |  60%
      |
      |++++++++++++++++++++++++++++++++                  |  64%
      |
      |++++++++++++++++++++++++++++++++++                |  68%
      |
      |++++++++++++++++++++++++++++++++++++              |  72%
      |
      |++++++++++++++++++++++++++++++++++++++            |  76%
      |
      |++++++++++++++++++++++++++++++++++++++++          |  80%
      |
      |++++++++++++++++++++++++++++++++++++++++++        |  84%
      |
      |++++++++++++++++++++++++++++++++++++++++++++      |  88%
      |
      |++++++++++++++++++++++++++++++++++++++++++++++    |  92%
      |
      |++++++++++++++++++++++++++++++++++++++++++++++++  |  96%
      |
      |++++++++++++++++++++++++++++++++++++++++++++++++++| 100%
    #>
      |
      |                                                  |   0%
      |
      |**                                                |   4%
      |
      |****                                              |   8%
      |
      |******                                            |  12%
      |
      |********                                          |  16%
      |
      |**********                                        |  20%
      |
      |************                                      |  24%
      |
      |**************                                    |  28%
      |
      |****************                                  |  32%
      |
      |******************                                |  36%
      |
      |********************                              |  40%
      |
      |**********************                            |  44%
      |
      |************************                          |  48%
      |
      |**************************                        |  52%
      |
      |****************************                      |  56%
      |
      |******************************                    |  60%
      |
      |********************************                  |  64%
      |
      |**********************************                |  68%
      |
      |************************************              |  72%
      |
      |**************************************            |  76%
      |
      |****************************************          |  80%
      |
      |******************************************        |  84%
      |
      |********************************************      |  88%
      |
      |**********************************************    |  92%
      |
      |************************************************  |  96%
      |
      |**************************************************| 100%
    #>
    #>
    #> *** Finished JAGS ***

## 3.3 The output from the fit

Here is the fitted spline curve to the number of unmarked fish available in each recovery stratum at the second trap

    demo.fit$plots$fit.plot

Fitted spline curve

The distribution of the posterior sample for the total number unmarked and total abundance that pass the second trap is available. Note this include the sum of the unmarked shown in the previous plot, plus a binomial distribution on the number of marked fish released that pass the second trap.

    demo.fit$plots$post.UNtot.plot

Distribution of posterior samples

A plot of the \(logit(P)\) is

    demo.fit$plots$logitP.plot

Estimates of logit(p)

In cases where there is little information, \(BTSPAS\) has shared information based on the distribution of catchability in the other strata.

A summary of the posterior for each parameter is also available. In particular, here are the summary statistics on the posterior sample for the total number unmarked and total abundance THAT PASS THE SECOND TRAP:

    demo.fit$summary[ row.names(demo.fit$summary) %in% c("Ntot","Utot"),]
    #>          mean       sd     2.5%     25%     50%     75%    97.5%     Rhat n.eff
    #> Ntot 22078.05 2266.638 17766.32 20557.0 22061.0 23548.0 26970.17 1.053470    44
    #> Utot 20288.79 2121.686 16318.37 18862.5 20263.5 21646.5 24956.76 1.052413    44

This also includes the Rubin-Brooks-Gelman statistic (\(Rhat\)) on mixing of the chains and the effective sample size of the posterior (after accounting for autocorrelation).

The estimated total abundance is 22,078 (SD 2,267 ) fish.

The estimated distribution function is allowed by vary by release stratum around a common “mean” distribution.

    probs <- demo.fit$summary[grepl("movep", row.names(demo.fit$summary)),  ]
    round(probs,3)
    #>           mean    sd  2.5%   25%   50%   75% 97.5%  Rhat n.eff
    #> movep[1] 0.130 0.029 0.081 0.109 0.127 0.148 0.196 1.001  1500
    #> movep[2] 0.513 0.054 0.406 0.478 0.514 0.549 0.618 1.006   350
    #> movep[3] 0.205 0.038 0.136 0.178 0.204 0.230 0.281 1.004   480
    #> movep[4] 0.085 0.023 0.046 0.068 0.083 0.098 0.135 1.002  1500
    #> movep[5] 0.037 0.014 0.014 0.028 0.036 0.046 0.070 1.000  1500
    #> movep[6] 0.012 0.008 0.002 0.007 0.011 0.016 0.029 1.006   350
    #> movep[7] 0.010 0.007 0.002 0.005 0.008 0.013 0.027 1.005  1300
    #> movep[8] 0.007 0.006 0.001 0.004 0.006 0.010 0.022 1.007   320

So we expect that about 13% of fish will migrate to the second trap in the day of release; about 51% of fish will migrate to the second trap in the second day after release etc.

The movement for each release stratum varies around this base distribution. It is also possible to see the probability of moving from release stratum \(i\) to recovery stratum \(j\) by looking at the \(Theta[i,j]\) values. Here are the transition probabilities for the first release stratum:

    round(demo.fit$summary[ grepl("Theta[1,", row.names(demo.fit$summary),fixed=TRUE),],3)
    #>             mean    sd  2.5%   25%   50%   75% 97.5%  Rhat n.eff
    #> Theta[1,1] 0.147 0.090 0.037 0.085 0.126 0.186 0.380 1.000  1500
    #> Theta[1,2] 0.491 0.140 0.209 0.396 0.493 0.588 0.747 1.004   650
    #> Theta[1,3] 0.204 0.096 0.057 0.129 0.192 0.262 0.426 1.002  1500
    #> Theta[1,4] 0.088 0.058 0.015 0.047 0.073 0.115 0.237 1.002  1100
    #> Theta[1,5] 0.039 0.031 0.004 0.018 0.031 0.052 0.115 1.000  1500
    #> Theta[1,6] 0.013 0.014 0.001 0.004 0.009 0.017 0.047 1.002  1100
    #> Theta[1,7] 0.010 0.011 0.000 0.003 0.007 0.013 0.041 1.004  1400
    #> Theta[1,8] 0.008 0.009 0.000 0.002 0.005 0.010 0.033 1.005   370

The probabilities should also sum to 1 for each release group.

As with the other non-parametric non-diagonal model, you can specify a prior distribution for the movement probabilities.

The sample of the posterior-distribution for the proportion of fish that DO NOT FALL back is

    round(demo.fit$summary[ grepl("ma.p", row.names(demo.fit$summary),fixed=TRUE),],3)
    #>   mean     sd   2.5%    25%    50%    75%  97.5%   Rhat  n.eff
    #>  0.614  0.057  0.498  0.576  0.614  0.653  0.722  1.049 47.000

It is always important to do model assessment before accepting the results from the model fit. Please contact me for details on how to interpret the goodness of fit, trace, and autocorrelation plots.

# 4 References

Bonner, S. J., & Schwarz, C. J. (2011). Smoothing population size estimates for Time-Stratified Mark–Recapture experiments Using Bayesian P-Splines. Biometrics, 67, 1498–1507. https://doi.org/10.1111/j.1541-0420.2011.01599.x

Schwarz, C. J. and Bonner, S. B. (2011). A spline-based capture-mark-recapture model applied to estimating the number of steelhead within the Bulkley River passing the Moricetown Canyon in 2001-2010. Prepared for the B.C. Ministry of Environment.

Schwarz, C. J., & Dempson, J. B. (1994). Mark-recapture estimation of a salmon smolt population. Biometrics, 50, 98–108.